Math Behind PMM

Token Type

base and quote are two concepts that will be mentioned frequently. Two easy ways to distinguish between them are:

  • In a trading pair, the base is always the token before the hyphen, and quote is after it

  • In transactions, the price refers to how many quote tokens are needed in exchange for one base token

For example, in the ETH-USDC trading pair, ETH is the base token and USDC is the quote token.

Pool Parameters

The funding pool of PMM is described by four parameters:

  • B0B_0: base token regression target - total number of base tokens deposited by liquidity providers

  • Q0Q_0: quote token regression target - total number of quote tokens deposited by liquidity providers

  • BB: base token balance - number of base tokens currently in the pool

  • QQ: quote token balance - number of quote tokens currently in the pool

PMM Formula

The PMM price curve is plotted by the following pricing formula:

Pmargin=iRP_{margin}=iR

Where RR is defined to be the piecewise function below:

SellBaseTokenif B<B0, R=1k+(B0B)2kSellBaseToken:if \ B<B_0, \ R=1-k+(\frac{B_0}{B})^2k

SellQuoteTokenif Q<Q0, R=1/(1k+(Q0Q)2k)Sell Quote Token:if \ Q<Q_0, \ R=1/(1-k+(\frac{Q_0}{Q})^2k)

else R=1else \ R=1

AMM & Stable Swap Use Cases

PMM supports transition to AMM and stable coin trading as well.

Traditional AMM

PMM is a generalization of AMM, when:

  • k = 1

  • deposit and withdraw in proportion to the current pool ratio

Stable swap

PMM can also be used as stable swap, when:

  • i = 1

  • k = 0.0001

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